Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation

For exponential last passage percolation on the plane we analyse the probability that the point-to-line geodesic exhibits an atypically large transversal fluctuation at the endpoint as well as the probability that the point-to-point geodesic exhibits an atypically large transversal fluctuation at the halfway point. In particular, we show that $p_n^{\ast }(t)$, the probability that the point-to-line geodesic from the origin to the line $x+y=2n$ ends at $(n-t(2n{)}^{2\mathrm{/}3},n+t(2n{)}^{2\mathrm{/}3})$ satisfies that $n^{2\mathrm{/}3}{p}_n^{\ast }(t)=\exp (-(\frac{4}{3}+o(1))t^3)$ for $t$ large and $p_{n,\frac{1}{2}}(t)$, the probability that the geodesic from the origin to the point $(n,n)$ passes through the point $(\frac{1}{2}n-t{n}^{2\mathrm{/}3},\frac{1}{2}n+t{n}^{2\mathrm{/}3})$, satisfies $n^{2\mathrm{/}3}{p}_{n,\frac{1}{2}}(t)=\exp (-(\frac{8}{3}+o(1))t^3)$ for $t$ large. The latter result solves a special case of a conjecture from Liu (PTRF, 2022).